Note: It's not really an alternative to Dropbox since you can't share your files with others. But personally I'm also using Dropbox to backup some of my important data (encrypted data like Bitcoin wallets or KeePass / 1Password file).
I was looking to setup a private incrementing encrypted backup, like rsync but with encryption. And I found duplicity. So I setup my cron to run this command every night:
$ duplicity --ssh-options="-oIdentityFile=~/.ssh/max.example.priv"
/home/max/wallets/ scp://max@example.com//home/max/backups/wallets/
Install it on OS X:
$ brew install duplicity
$ pip install paramiko
On Ubuntu
$ sudo apt-get install duplicity